# Do all massless particles (e.g. photon, graviton, gluon) necessarily have the same speed $c$?

I suppose there was a discussion already on speed-of-gravity-and-speed-of-light.

But I silly wonder whether all the massless mediators of four fundamental forces, i.e.

Graviton: $g_{\mu\nu}$ (gravity)

Photon $\gamma$: $A_\mu$ (electromagnetism)

Gluons: $A_\mu^a$ (strong interactions)

Necessarily travel at the same speed? Is there a no-go theorem or theoretical proof to say that it is impossible to have these three mediators have different speeds?

Or does QCD confinement makes the story of gluons any different from gravitons and photons?

[ps. excluded massive $Z^{0}$ and $W^{\pm}$ bosons (weak interactions)]

Another way to say this: Speed of photon, graviton, gluon all equal to $c$? or Whether all massless particles necessarily have the same speed?

Note add: Notice however, in condensed matter systems, there can be emergent gauge fields and emergent massless particles (Dirac or Weyl cones), but their speeds need not be the same, unless there is some emergent symmetry...

• Isn't it because of the geometry of spacetime? – unsym Dec 17 '13 at 5:41
• Can we have bimetric gravity with, say, photons and gravitons coupled to different metrics? – user23660 Dec 17 '13 at 5:50
• @hwlau: can you be more explicit? Actually now I wonder whether one scheme is that the SUSY unification on gauge couplings of U(1)xSU(2)xSU(3) and gravity at the GUT scale 10^16 GeV, implies that the mediators of all forces needs to have the same origin, so the speed of massless particles needs to be the same? What may be a theoretical constraint? or a no-go theorem? – wonderich Dec 17 '13 at 5:52
• ps. If $c_{gravity}, c_{photon}$ and $c_{gluon}$ are different, we can imagine there are more fundamental constants, and HEP people certainly hate to see this happens. – wonderich Dec 17 '13 at 5:58
• There was a theory which predicted different limit speeds for each particle, but I can find the reference. – jinawee Dec 17 '13 at 7:44

If basic symmetry and homogeneity assumptions about the Universe hold, then yes, all massless real particles (see Anna V's answer for virtual particles must travel at a universal constant $c$, the speed of a massless particle, in all frames of reference.

Given these basic symmetry and homogeneity assumptions, one can derive the possible co-ordinate transformations for the relativity of inertial frames: see the section "From Group Postulates" on the Wikipedia Page "Lorentz Transformation". (Also see my summary here). Galilean relativity is consistent with these assumptions, but not uniquely so: the other possibility is that there is some speed $c$ characterizing relativity such that $c$ is the same when measured from all frames of reference. Time dilation, Lorentz-Fitzgerald contraction and the impossibility of accelerating a massive particle to $c$ are all simple consequences of these other possible relativities.

So now it becomes an experimental question as to which relativity holds: Galilean or Lorentz transformation? And the experiment is answered by testing how speeds transform between inertial frames. Otherwise put, the experimental question is are there any speeds that are the same for all inertial observers?. The question is not about measuring the values of any speed, but rather, how they transform. Now of course we know the answer: the Michelson Morley experiment found such a speed, the speed of light. So there are two conclusions here: (1) Relativity of inertial frames is Lorentzian, not Galilean (which can be thought of as a Lorentz transformation with infinite $c$) and (2) light is a massless particle, because light is observed to go at this speed that transforms in this special way.

Notice that at the outset of this argument we mention nothing about particles or any particular physical phenomenon (even though special relativity's historical roots were in light). It follows that, if $c$ is experimentally observed to be finite (i.e. Galilean relativity does not hold), then the specially invariant speed is unique: it can only be reached by massless particles and there can't be more than one such $c$ - the Lorentz laws are what they are and are the only ones consistent with our initial symmetry and homogeneity assumptions. So if we observed two different speeds transforming like $c$, this would falsify our basic symmetry and homogeneity assumptions about the World. No experiment gives us grounds for doing that.

This is why all massless particles have the same speed $c$.

Update: Experimental Results

As is now common knowledge, the gravitational wave event GW170817 and gamma ray burst GRB170817A give strong experimental evidence of the equality of the speeds of light and gravitation. As discussed in:

Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A

the 1.7 second time delay between the gravitational wave arrival and the gamma ray burst, together with conservative assumptions about other sources of delay, yields an experimental bound on the fractional difference between the speed of light and of gravitation:

$$\frac{v_g-v_{em}}{c} \leq 3\times 10^{-15}$$

an impressive experimental bound indeed. Within the next 10 years, we probably shall see several such events, and thus this experimental bound will tighten further (unless something really theoretically unforeseen happens!).

Mass From Confined Massless Particles

Incidentally, if we confine massless particles, e.g. put light into a perfectly reflecting box, the box's inertia increases by $E/c^2$, where $E$ is the energy content. This is the mechanism for most of your body's mass: massless gluons are confined and are accelerating backwards and forwards all the time, so they have inertia just as the confined light in a box did. Likewise, an electron can be thought of as comprising two massless particles, tethered together by a coupling term that is the mass of the electron. The Dirac and Maxwell equations can be written in the same form: the left and right hand circularly polarized components of light are uncoupled and therefore travel at $c$, but the massless left and right hand circular components of the electron are tethered together. This begets the phenomenon of the Zitterbewegung - whereby an electron can be construed as observable at any instant in time as traveling at $c$, but it swiftly oscillates back and forth between left and right hand states and is thus confined in one place. Therefore it takes on mass, just as the "tethered" light in the box does.

• I like your answer about the origin of rest mass, if I may put it that way. Do you have some references that go deeper into this? Ive always found it curious that there should be two origins of inertia; the discussed here, and the higgs field. Or is there some perspective which I am missing under which these are two different ways of looking at the same mechanism? – Eelco Hoogendoorn Jul 2 '14 at 20:20
• @EelcoHoogendoorn THanks: as I understand it (I'm not a particle physicist) the Higgs mechanism can also be understood as a coupling between the weak bosons and the Higgs field. If you can find a copy of Penrose's "Road to Reality" look at the way he described the Zitterbewegung of the electron. The coupling preventing the weak bosons zipping off a lightspeed is the rest mass. The origin of rest mass in the way I describe is an old idea: it goes right back to Einstein's $E=m\,c^2$ paper .... – WetSavannaAnimal Jul 3 '14 at 12:48
• @EelcoHoogendoorn ...."Does the inertia of a body depend upon its energy content?". I take this kind of idea up here – WetSavannaAnimal Jul 3 '14 at 12:56
• Right. I see how binding energy leads to mass, and how this nicely obeys all relativistic notions. This follows simply from two particles bound by a retarded potential. This has always strongly suggested to me that the rest mass of supposed elementary particles might also be due to the binding of some substructure. The standard counter to that is that such a substructure does not exist, and Higgs is needed. You say that the binding between the components of the zitterbewegung would explain the mass of the electron. Are these mere different perspectives, or should that Nobel be recalled? – Eelco Hoogendoorn Jul 5 '14 at 11:15

Another way to say this: Speed of photon, graviton, gluon all equal to c? or Whether all massless particles necessarily have the same speed?

You must not have been introduced to the concept of a virtual particle:

In physics, a virtual particle is a transient fluctuation that exhibits many of the characteristics of an ordinary particle, but that exists for a limited time. The concept of virtual particles arises in perturbation theory of quantum field theory where interactions between ordinary particles are described in terms of exchanges of virtual particles. Any process involving virtual particles admits a schematic representation known as a Feynman diagram, in which virtual particles are represented by internal lines.

A virtual particle is an internal line in a Feynman diagram which represents the propagator mathematics that has to be substituted to get the integral necessary for computing measurable quantities . Virtual particles have the quantum numbers of their homonymous ( having the same name) particles except not the mass. The mass is off shell.

So it is a general rule that massless particles travel at the velocity of light, but only when in external lines in Feynman diagrams. This is true for photons, and we thought it was true for neutrinos but were proven wrong with neutrino oscillations.

Gluons on the other hand we only find within a nucleus and these are by definition internal lines in Feynman diagrams and therefore are not constrained to have a mass of 0, even though in the theory they are supposed to. In the asymptotically free case, at very high energies they should display a mass of zero.

• The path integral of physical fields and virtual particles...I feel like it's a bit like we consider the exponetial function $\mathrm e^x$, make a tailor expansion around $\pi$ up to second order and absorb an overall factor $\mathrm e^{-\pi}/2$ because it's not physically detactable. We end up with $\mathrm e^{x}\ \overset{\text{sort of}}{=}\ 2+\pi^2-2\pi(1+x)+x(2+x)$. We absorb the constant and reparametrize $ex\equiv x+1$ to emphasise the beautiful symmetry of our system $-2\pi\ ex+(ex-1)(ex+1)$. We call $ex$ the "expandion" and it will provide great counterexamples for other somethingions. – Nikolaj-K Dec 17 '13 at 10:29
• I think Nick's comment is a serious concern. At least in QED it seems like this perturbative calculation makes sense (weak coupling), at least for a while. However, the series is asymptotic. In QCD we have an even bigger problem with strong coupling. So while the mathematics is useful and suggests that there are virtual particles, I think the onotlogical status of such objects hasn't really been established. Are they physical or just an artifact of the perturbative technique? – Kevin Driscoll Dec 17 '13 at 10:41
• @KevinDriscoll Well, when really kicked out of the nucleus gluons do make jets so their reality seems to me to be backed by experiment.cerncourier.com/cws/article/cern/29201 – anna v Dec 17 '13 at 12:39
• @NickKidman I think the difference with your example lies in the quantum numbers . The virtual particles are not arbitrary functions, they carry the quantum number conservations and the pole/propagetor that represents them has the on mass shell value except that in the internal lines things go imaginary. – anna v Dec 17 '13 at 12:41
• @annav: Are you implying the expandions are arbitary? While $x$ might seem like a random variable, only the expandions let us explore the quadrahedron density $\mathcal L_*[ex]:=(ex-1)(ex+1)$. Note that this purely algebraic (=geometric) object is invariant under $ex\mapsto -\mathrm{wau}\cdot ex$, (Ref 17). Indeed, the symmetry group is isomorphic to the Galois group which emerges when you pass from the unique Archimedean complete totally ordered field to its algebraic closure, (Ref 32). – Nikolaj-K Dec 17 '13 at 13:23

It is not hard to imagine a toy universe in which different fundamental forces propagate at different speeds. However, a necessary consequence of that would be violations of lorentzian symmetry, and the ability to triangulate a preferred rest frame.

Although I don't see a theoretical reason why these speeds need be the same (I might be missing something though, perhaps some stability arguments require it), empirically there isn't much room for different speeds.

I thought the same thing for a long time. I wondered why gluons don't fly out of the nucleus at the speed of $c$. The difference is that photons don't interact with other photons and gravitons don't interact with other gravitons. They can move around and pass through each other. On the other hand, gluons do interact with each other.

In fact, gluons form chains/flux tubes which is part of why quarks are confined. Gluons do travel at $c$ but not for very far before they interact with other quarks or gluons, which keeps them from moving any appreciable distance.

• no. this is not true? gravitons DO interacting with gravitons. gravity are badly non-renormalizable. – wonderich Dec 17 '13 at 5:40
• @Idear I'm pretty sure gravitons behave normally in one-loop diagrams and don't interact very strongly with each other when they do interact. The strong force, on the other hand, is many orders of magnitude stronger in interaction. – Brandon Enright Dec 17 '13 at 5:43
• Thanks Brandon. What you address is not exactly what I asked. I wonder whether there is no go theorem for preventing the speeds of massless particles to be different. Perhaps if the speed of one is the largest, say c_1 >c_2 >c_3> .... Then only c_1 can be defined as massless? – wonderich Dec 17 '13 at 5:44
• @Idear graviton-related interaction question physics.stackexchange.com/questions/8112/… – Brandon Enright Dec 17 '13 at 5:54
• @Idear John Rennie also addresses strong-force propagation velocity via Gluons and Mesons here: physics.stackexchange.com/questions/57137/… – Brandon Enright Dec 17 '13 at 5:59

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